It may be worth adding that the planetary ephemeris are computed by NASA (JPL) by integrating the equations of motion forward in time numerically. Then the res

It may be worth adding that the planetary ephemeris are computed by NASA (JPL) by integrating the equations of motion forward in time numerically. Then the results (e.g. planet centroid as a function of time) are encoded in a file, as noted above. That file is then read by the ephemeris software to obtain the position at the desired time. I haven’t checked in a few years, but there used to be multiple files to choose from depending on the time range you needed to cover. It’s free — “your taxpayer dollars at work.”